Note: This page uses SAS 9.2.
The tobit model, also called a censored regression model, is designed to estimate linear relationships between variables when there is either left- or right-censoring in the dependent variable (also known as censoring from below and above, respectively). Censoring from above takes place when cases with a value at or above some threshold, all take on the value of that threshold, so that the true value might be equal to the threshold, but it might also be higher. In the case of censoring from below, values those that fall at or below some threshold are censored.
Please note: The purpose of this page is to show how to use various data analysis commands. It does not cover all aspects of the research process which researchers are expected to do. In particular, it does not cover data cleaning and checking, verification of assumptions, model diagnostics and potential follow-up analyses.
Example 1. In the 1980s there was a federal law restricting speedometer readings to no more than 85 mph. So if you wanted to try and predict a vehicle's top-speed from a combination of horse-power and engine size, you would get a reading no higher than 85, regardless of how fast the vehicle was really traveling. This is a classic case of right-censoring (censoring from above) of the data. The only thing we are certain of is that those vehicles were traveling at least 85 mph.
Example 2. A research project is studying the level of lead in home drinking water as a function of the age of a house and family income. The water testing kit cannot detect lead concentrations below 5 parts per billion (ppb). The EPA considers levels above 15 ppb to be dangerous. These data are an example of left-censoring (censoring from below).
Example 3. Consider the situation in which we have a measure of academic aptitude (scaled 200-800) which we want to model using reading and math test scores, as well as, the type of program the student is enrolled in (academic, general, or vocational). The problem here is that students who answer all questions on the academic aptitude test correctly receive a score of 800, even though it is likely that these students are not "truly" equal in aptitude. The same is true of students who answer all of the questions incorrectly. All such students would have a score of 200, although they may not all be of equal aptitude.
Let's pursue Example 3 from above. We have a hypothetical data file, tobit.sas7bdat with 200 observations with format defined below. The academic aptitude variable is apt, the reading and math test scores are read and math respectively. The variable prog is the type of program the student is in, it is a categorical (nominal) variable that takes on three values, academic (prog = 1), general (prog = 2), and vocational (prog = 3). Variable prog comes with a format provided below.
proc format; value proga 1="academic"; 2="general"; 3="vocational"; run;
Let's look at the data. Note that in this dataset, the lowest value of apt is 352. Note that no students received a score of 200 (i.e., the lowest score possible), meaning that even though censoring from below was possible, it does not occur in the dataset.
options nolabel nocenter nodate formchar = '|----|+|---+=|-/<>*'; proc means data = tobit maxdec=2 nonobs; class prog; vars apt read math; run; prog Variable N Mean Std Dev Minimum Maximum --------------------------------------------------------------------------------------------- academic apt 45 639.02 78.63 454.00 800.00 read 45 49.76 9.23 28.00 68.00 math 45 50.02 7.44 35.00 63.00 general apt 105 677.76 88.21 462.00 800.00 read 105 56.16 9.59 34.00 76.00 math 105 56.73 8.73 38.00 75.00 vocational apt 50 561.72 92.76 352.00 800.00 read 50 46.20 8.91 31.00 68.00 math 50 46.42 7.95 33.00 75.00 --------------------------------------------------------------------------------------------- ods graphics / reset=all imagename='dens' imagefmt=png width=4in height=4in border=off; proc sgplot data = tobit noautolegend; histogram apt; density apt /type = normal lineattrs=(color=blue); run;
Looking at the above histogram showing the distribution of apt, we can see the censoring in the data, that is, there are far more cases with scores of 775 to 800 (i.e., the final bin) than one would expect looking at the rest of the distribution. Below is an alternative histogram that further highlights the excess of cases where apt=800. In the histogram below, midpoints option is used to produce a histogram where each unique value of apt has its own bar by specifying that there should be bins from 350 (the minimum of apt is 352) and a max of 800 in units of 1. Because apt is continuous, most values of apt are unique in the dataset, although close to the center of the distribution there are a few values of apt that have two or three cases. The spike on the far right of the histogram is the bar for cases where apt=800, the height of this bar relative to all the others clearly shows the excess number of cases with this value.
Next we'll explore the bivariate relationships in our dataset. We make use of the scatter matrix plot created by proc corr via ods graphics on option.ods graphics / reset=all imagename='hist' imagefmt=png width=4in height=4in border=off; proc univariate data=tobit noprint; histogram apt / midpoints=350 to 800 by 1 normal ; run;
ods graphics / reset=all imagename='mat' imagefmt=png width=4in height=4in border=off; ods graphics on; proc corr data = tobit nosimple; var read math apt; run; ods graphics off; Pearson Correlation Coefficients, N = 200 Prob > |r| under H0: Rho=0 read math apt read 1.00000 0.66228 0.64512 <.0001 <.0001 math 0.66228 1.00000 0.73327 <.0001 <.0001 apt 0.64512 0.7332
Note the collection of cases at the top of the bottom row of the scatter plots are due to the censoring in the distribution of apt.
Below is a list of some analysis methods you may have encountered. Some of the methods listed are quite reasonable while others have either fallen out of favor or have limitations.
Below we use proc qlim to fit a tobit regression model. Note that proc qlim is part of the ETS module for SAS. It is also possible to fit a tobit model using proc lifereg (part of the STAT module), although the syntax to do so is somewhat different from the example shown below. The class statement identifies prog as a categorical variable, and the model statement specifies that apt should be modeled using read, math, and prog. The endogenous statement specifies that the outcome variable apt is censored, with an upper bound of 800 (i.e., ub=800).
proc qlim data = tobit ; class prog; model apt = read math prog; endogenous apt ~ censored (ub=800); run;< The QLIM Procedure Summary Statistics of Continuous Responses N Obs N Obs Standard Lower Upper Lower Upper Variable Mean Error Type Bound Bound Bound Bound apt 640.035 99.219030 Censored 800 17 Class Level Information Class Levels Values prog 3 academic general vocational Model Fit Summary Number of Endogenous Variables 1 Endogenous Variable apt Number of Observations 200 Log Likelihood -1041 Maximum Absolute Gradient 8.40561E-7 Number of Iterations 26 Optimization Method Quasi-Newton AIC 2094 Schwarz Criterion 2114
Parameter Estimates Standard Approx Parameter DF Estimate Error t Value Pr > |t| Intercept 1 163.422155 30.408580 5.37 <.0001 read 1 2.697939 0.618806 4.36 <.0001 math 1 5.914484 0.709818 8.33 <.0001 prog academic 1 46.143900 13.724195 3.36 0.0008 prog general 1 33.429162 12.955628 2.58 0.0099 prog vocational 0 0 . . . _Sigma 1 65.676720 3.481423 18.86 <.0001
Under the heading Parameter Estimates we see the coefficients, their standard errors, the t-statistics, and associated p-values. The coefficients for read and math are statistically significant, as are the terms for prog="academic" and prog="general" (with prog="vocational" as the reference category). Tobit regression coefficients are interpreted in the similiar manner to OLS regression coefficients; however, the linear effect is on the uncensored latent variable, not the observed outcome. See (McDonald 1980) for more details.
proc qlim data = tobit outest=t; class prog; model apt = read math prog; endogenous apt ~ censored (ub=800); run; proc print data = t noobs; run; prog_ prog_ prog_ _NAME_ _TYPE_ _STATUS_ Intercept read math academic general vocational _Sigma PARM 0 Converged 163.422 2.69794 5.91448 46.1439 33.4292 . 65.6767 STD 0 Converged 30.409 0.61881 0.70982 13.7242 12.9556 . 3.4814 proc qlim data = tobit ; class prog; model apt = read math prog; endogenous apt ~ censored (ub=800); test 'prog' prog_academic=0, prog_general =0; run;
Because the model is the same, the output for this syntax is the same as before, except for the addition section shown showing the results of the test statement. Under Test Results, we see that the overall effect of prog is statistically significant.
We can also test additional hypotheses about the differences in the coefficients for different levels of prog. Below we test that the coefficient for prog="academic" is equal to the coefficient for prog="general".
proc qlim data = tobit; class prog; model apt = read math prog; endogenous apt ~ censored (ub=800); test 'academic vs. general' prog_academic - prog_general = 0; run;
We may also wish to evaluate how well our model fits. This can be particularly useful when comparing competing models. One method of assessing model fit is to compare the predicted values based on the tobit model to the observed values in the dataset. Below we use proc qlim to generate predicted values along with the data via the output statement. Then proc corr is used to estimate the correlation between the predicted and observed values of apt. The output from proc corr gives the correlation between the predicted and observed values of apt, which is 0.78094. If we square this value, we get the squared multiple correlation, this indicates that the predicted values share about 61% (0.78094^2 = .6099) of their variance with the observed values of apt.
proc qlim data=tobit ; model apt = read math prog; endogenous apt ~ censored (ub=800); output out = temp1 predicted; run; proc corr data = temp1 nosimple; var apt p_apt; run; Pearson Correlation Coefficients, N = 200 Prob > |r| under H0: Rho=0 apt P_apt apt 1.00000 0.78094 <.0001 P_apt 0.78094 1.00000 <.0001
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